Simulated nucleon–nucleon and nucleon–nucleus reactions in the frame of the cascade exciton model at high and intermediate energies

P

— journal of April 2015

physics pp. 581–590

Simulated nucleon–nucleon and nucleon–nucleus reactions in the frame of the cascade exciton model

at high and intermediate energies

A ABDEL-HAFIEZ^1,∗, SHAKER EL-SHATER²and M F ZAKI¹

1Experimental Nuclear Physics Department, NRC, Atomic Energy Authority, Cairo, Egypt

2Physics Department, Faculty of Science, El-Bath University, Homs, Syria

∗Corresponding author. E-mail: abdel_hafiez@yahoo.com

MS received 22 February 2014; revised 2 May 2014; accepted 13 May 2014 DOI: 10.1007/s12043-014-0852-0; ePublication: 26 February 2015

Abstract. In this study, we have used the cascade exciton model (CEM) to investigate different characteristics of nuclear reactions. Number of nucleon–nucleon collisions in Pb+Pb collisions as a function of impact parameter and rapidity distributions of negative particles from p+Ar and p+Xe interactions atp_lab = 200 GeV/c have been studied. We could create inclusive spectra of pions for separate charged states from reactions and total neutron multiplicities per primary reaction at 1000 MeV for different thin targets. Also, cross-sections for the reactions²⁰⁹Bi(p, f) and²⁰⁹Bi(n, f) were studied. Interactions of 1.0 GeV protons with C, Al, Cu, Sn, and Pb are presented in this study.

All the calculated characteristics are compared with other theoretical calculations and compared with the experimental data. CEM shows good agreement with both theoretical and experimental results. In this study, we have used quantum molecular dynamic (QMD) as a theoretical model to compare our results.

Keywords. Nuclear reactions; quantum dynamics; cascade exciton model.

PACS No. 24

1. Introduction

The concept of the intranuclear cascade model is quite old and intuitively simple. A particle incident on a nucleus will interact with individual nucleons, with the final states defined by a set of fundamental particle–particle cross-sections. The nucleons are con- sidered to be cold free gas confined within a potential that describes nuclear density as a function of radius, and the Fermi motion of the nucleons is taken into account in modelling the interactions [1].

Recently, the cascade exciton model (CEM) has been used extensively by taking into account the competition between particle emission and fission at the compound nucleus stage [2] and a more realistic calculation of nuclear level density [3]. Earlier, an extended

version of the CEM, as realized in the code CEM92, was used by Konshin [4] to cal- culate nucleon-induced fission cross-sections for actinides in the energy region between 100 MeV and 1 GeV. Nevertheless, the CEM has not been applied previously to study fission cross-sections for pre-actinides. Therefore, its predictive power and applicability to evaluate arbitrary fission cross-sections is unknown.

Cascade exciton model (CEM) of nuclear reactions was initially proposed to describe nucleon-induced reactions at bombarding energies below or at∼100 MeV, but lately this is used even for a larger interval of incident nucleon and pion bombarding energies (see, e.g., [5] and references given therein).

The aim of this paper is to obtain and analyse the relativistic kinetic equations describ- ing different stages of nuclear reactions in the CEM, to show the main assumptions of our model and to demonstrate the exemplary results obtained using CEM.

2. Theory and the main concepts of the CEM model

In the CEM, it is assumed that the reactions occur in three stages. The first stage is the intranuclear cascade in which primary particles can be rescattered several times prior to absorption by, or escape from, the nucleus. The excited residual nucleus, after the emis- sion of the cascade particles, determines the particle–hole configuration that is the starting point for the second, pre-equilibrium stage of the reaction. The subsequent relaxation of the nuclear excitation is treated in terms of the exciton model of pre-equilibrium decay that includes a description of the equilibrium evaporative third stage of the reaction [6]. In general, the three components may contribute to any experimentally measured quantity, in particular for the inclusive particle spectrum

σ(p)dp=σ_in

N^cas(p)+N^peq(p)+N^eq(p)

dp. (1)

The inelastic cross-sectionσ_in is not taken from the experimental data or the indepen- dent optical model calculations, rather it is calculated within the cascade model itself [7].

Hence, the CEM predicts absolute values for the calculated characteristics and does not require any additional data or special normalization of its results. The CEM allows neu- trons and protons upto 5 GeV and pions upto 2.5 GeV to initiate nuclear reactions. Valid targets are nuclei with charge greater than 5 and a mass number greater than 11. The CEM consists of an intranuclear cascade model, followed by a pre-equilibrium model and an evaporation model.

N^cas(p)dp= 1 σ_in

R 0

d²b

r>R

dr tcas

0

dt f_b^cas(r, p, t )dp, (2) where the integration is carried out over all accessible impact parametersbfor particles leaving a nucleus of radius R by the end of the cascade staget_cas.f^cas (r, p, t) is the single-particle distribution function through which all the characteristics needed can be expressed.

All the cascade calculations are carried out in a three-dimensional geometry. The nuclear matter densityρ(r) is described by a Fermi distribution with two parameters taken from the analysis of electron–nucleus scattering, namely

ρ (r)=ρ_p(r)+ρ_n(r)=ρ₀

1+exp

(r−c)/a

, (3)

wherec=1.07A^1/3fm,Ais the mass number of the target anda=0.545 fm.

The interaction of the incident particle with the nucleus results in a series of successive quasifree collisions of the fast cascade particles (π or N) with intranuclear nucleons:

N N→N N, N N →π N N, N N→π₁, . . . , πiN N

π N→π N, π N→π₁, . . . , π_iN (i≥2) . (4) To describe these elementary collisions, one uses the experimental cross-sections for the freeπ NandN N interactions approximated by special polynomial expressions with energy-dependent coefficients [8] and one takes into account the Pauli principle.

Besides the elementary processes (4), the Dubna intranuclear cascade model (INC) also takes into account pion absorption on nucleon pairs

π+[N N]→N N. (5)

The momenta of two nucleons participating in the absorption are chosen randomly from the Fermi distribution, and the pion energy is distributed equally between these nucleons in the centre-of-mass system of the pion and nucleons participating in the absorption.

The direction of motion of the resultant nucleons in this system is taken to be isotropi- cally distributed in space. The effective cross-section for absorption is estimated from the experimental cross-section for pion absorption by deuterons

σ (π⁻+np→nn)=W·σ (π⁻+d→nn), (6) whereW is the variation of the absolute normalization of pion absorption cross-section.

The quantityW depends on the pion energyT_π, the characteristics of the target nucleus, the point where the pion is absorbed, and on the spin–isospin states of absorbing pairs [9].

The kinetic energy can be written in the form of a master equation as

∂P (E, α, t )

∂t =

α=α

λ (Eα, E_α) P

E, α, t −λ (Eα, E_α) P (E, α, t ) . (7) HereP (E, α, t )is the probability of finding the system at a time momenttin theEαstate andλ(Eα,Eα)is the energy-conserving probability rate, defined in the first order of the time-depending perturbation theory

λ (Eα, Eα)=2π

¯

h | Eα| ˆV|Eα|²ωα(E). (8) The matrix element Eα| ˆV|Eαis believed to be a smooth function in energy andωα(E) is the density of the final state of the system.

The pre-equilibrium component in (1) for the inclusive spectrum of particles having momentum within the interval dp around the valuepcan be represented as

N^peq(p)dp= teq

t_cas

dt

n

λ (n, E, T ) P (E, n, t )∂ (p, )

∂ (T , )F ( )dT d (9)

the angular functionF ( )is normalized to unity,

d F ( )=1. We can write down the expression for the equilibrium (n≥n_eq)component as:

N^eq(p)dp= _∞

teq

dt

n

λ (n, E, T ) P (E, n, T )∂ (p, )

∂ (T , )F ( )dT d , (10) where the time momentt → ∞corresponds to the complete de-excitation of a nucleus due to particle emission. As the nuclear states with differentnare equally probable in statistical equilibrium, the right-hand side of (10) is simplified to

N^eq(p)dp=

n

λ (n, E, T ) P (E, n, t )ω (n−1, E−B,−T )

n^tω (n^t, E)

×∂ (p, )

∂ (T , )F ( )dTd

∼T^3/2σin(T )

nω (n−1, E−B,−T )

n^tω (n^t, E) dTd , (11)

whereω(E) is the total density of the excited states.

3. Results

In nucleus–nucleus collisions, for a given impact parameterbwe can deduce the number of nucleon–nucleon collisions,N_col. It is given by the following parametrized function [10]:

N_col=N₀exp

− b

b₀ 2

−a₀b, (12)

where N₀ = 1550, b₀ = 7 fm anda₀ = 0.5 for Pb+Pb collisions (see figure 1) and N₀ =180,b₀ =4.5 fm anda₀ =0 for Ca+Ca collisions. Resonance production cross- sections as well as the contribution to the background from openbandcproduction are estimated fromN_col, wherebis the impact parameter andcis given in eq. (3). The most important variable in ion–ion collisions is the multiplicity of a given impact parameter which is calculated using the following functions for Pb+Pb collisions:

dN^↑±

/dy

↓(y=0)

=1.18(1−(b/11.5))

d

N^↑±_↑ /dy

↓

(y=0)^↑(b=0) ifb <10 fm

dN^↑±

/dy

↓(y=0)

=5.9 exp(2−0.56463b)

d

N^↑±_↑ /dy

↓

(y=0)^↑(b=0) (13) ifb >10 fm

whereyis the rapidity andN^±is the multiplicity range.

Figure 1. The number of nucleon–nucleon collisions in Pb+Pb collisions as a function of impact parameter.

The general parameter of Pb beam used in this paper is taken from [10].

In figure 2 the typical inclusive rapidity distributions of negative particles from p+Ar and p+Xe collisions atp_lab=200 GeV/c are displayed [11]. Our CEM calculations are compared with both the experimental data and the QMD model calculations. There are

Figure 2. Rapidity distributions of negative particles from p+Ar and p+Xe interac- tions atp_lab = 200 GeV/c and theoretical predictions. Closed circles with error bars are experimental data from [11]. Solid histograms are the CEM calculations and dotted histograms are the QMD model calculations.

Figure 3. Inclusive spectra of pions for separate charged states from reactions are indicated. Points are experimental data from [12]. The solid histograms are the CEM calculations. Double differential cross-sections at angles 30^◦, 60^◦, 80^◦and 120^◦ at 562.5 MeV neutrons on Cu.

no data for heavy nuclei at higher energies. One concludes from figure 2 that these mea- surements of non-identified secondaries are of rather modest precision and that theoretical calculations, based on a string model and including multiple interactions in the Glauber framework, are close to the data.

Figure 4. Total neutron multiplicities of the first reaction at 1000 MeV for different thin targets. The calculations were done using the QMD and CEM models. Data are taken from [13].

Table 1. The parameters of the nucleon-induced fission cross-section approximations for²⁰⁹Bi.

E (MeV) Expression C₀ C₁ C₂

209Bi(p, f)

37.4–70 (14a) −5.150 0.2203 −0.001290

70–3000 (14c) 182.7 0.01264 43.60

209Bi(n, f)

20–73 (14b) −108.7 50 −5.6

73–1000 (14c) 100 0.006 45

In figure 3 CEM calculations are applied to analyse practically all known data of nucleon-induced pion productions for intermediate and heavy nuclei and bombarding energies less than several GeV.

In figure 4 the neutron multiplicities calculated in reactions induced by protons on Bi, Au, Pb, W, Th, Hg, U, Fe and Cu thin targets are compared with experimental data in the energy range 1–1.2 GeV. The calculations were done using the QMD and CEM models.

The data were taken from [13].

Figure 5. Comparison between the experimental data and the calculations of the cross-sections for the following reactions: (a)²⁰⁹Bi(p, f) and (b)²⁰⁹Bi(n, f). The solid lines represent the approximations of experimental data given in table 1, the dashed lines show the CEM predictions.

Figure 6. Experimental spectra of (a) neutrons and (b) protons emitted at 140^◦from the interactions of 1.0 GeV protons with C, Al, Cu, Sn and Pb [14] compared with the CEM (solid histogram) and QMD models calculations (dashed histograms) [15].

The experimental data on the nucleon-induced fission cross-sections for²⁰⁹Bi together with the corresponding approximations are described by the following expressions:

σ_f(E)=exp

c₀+c₁E+c₂E² , (14a)

σ_f(E)=exp

c₀+c₁lnE+c₂(lnE)² , (14b)

σ_f(E)=c₀{1−exp [−c₂(E−c₂)]}, (14c) whereσ_fis the fission cross-section,Eis the incident particle energy andc₀,c₁andc₂are fitting parameters found by the least-square method and given in table 1.

The nucleon-induced fission cross-section calculations for²⁰⁹Bi nuclei at 45–500 MeV performed in this work with the CEM code show that there is no universal input parameter set that allows us to describe the fission excitation functions in the whole energy region considered. In the lower energy part (up to 160 MeV) the best agreement with the exper- imental data is reached. Figure 5 illustrates the results of the calculations with the given parameters in comparison with the experimental data approximations.

In figure 6 the calculations of CEM are compared with the experimental spectra of neutrons and protons emitted at 140^◦from the interaction of 1 GeV protons [16]. One can see that for medium and light nuclei, the results obtained usingW =2 agree much better with the data compared with the standard CEM results(W =4). From these and similar results obtained for other targets at other intermediate energies, we can conclude thatW seems to depend on the atomic number of the target; increasing from about 2 for light targets like C to about 4 for heavy targets like Pb, whereW is the variation of absolute normalization of the absorption cross-section defined in eq. (6).

4. Summary

In this paper, we have used the CEM for calculating many characteristics of various nuclear reactions. These results have been compared with the experimental data and also with other theoretical calculations. Here, we have used QMD model for theoretical cal- culations. As we could see from these results, CEM gives a good agreement with the others.

The CEM can describe the absolute value of various characteristics of pion- and nucleon-induced reactions for medium and heavy target nuclei and incident energies less than several GeV. Also, we have performed the analysis of the nucleon-induced fission cross-section for Bi nuclei in the 45–500 MeV energy range.

Acknowledgement

The authors would like to thank Dr Salem Ba-Fatoum, the College Dean, Faculty of Edu- cation at Almahra, Hadramout University, Yemen Republic for his guidance and fruitful assistance to accomplish this work.

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